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Source-Dependence of Utility and Loss Aversion

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Source-Dependence of Utility and Loss Aversion Source‐dependence of utility and loss aversion: A critical test of ambiguity models Mohammed Abdellaoui HEC‐Paris & GREGHEC‐CNRS, [email protected] Han Bleichrodt* Erasmus School of Economics, Rotterdam, [email protected] Olivier l’Haridon Crem‐Université de Rennes 1 & GREGHEC, olivier.lharidon@univ‐rennes1.fr Dennie van Dolder Erasmus School of Economics, Rotterdam, [email protected] Abstract: This paper tests whether utility is the same for risk and for uncertainty. This test is critical to distinguish models that capture ambiguity aversion through a difference in utility between risk and uncertainty (like the smooth ambiguity model) and models that capture ambiguity aversion through a difference in event weighting between risk and uncertainty (like multiple priors and prospect theory). We designed a new method to measure utility and loss aversion under uncertainty without the need to introduce simplifying parametric assumptions. Our method extends Wakker and Deneffe’s (1996) trade‐off method by allowing for standard sequences that include gains, losses, and the reference point. It provides an efficient way to measure loss aversion and a useful tool for practical applications of ambiguity models. We could not reject the hypothesis that utility and loss aversion were the same for risk and uncertainty suggesting that utility reflects attitudes towards outcomes and not attitudes towards ambiguity. Sign‐dependence was important both for risk and for uncertainty. Utility was S‐shaped, concave for gains and convex for losses and there was substantial loss aversion. Our findings support models that explain ambiguity aversion through a difference in event weighting and suggest that descriptive models of ambiguity aversion should allow for reference‐dependence. JEL: C91, D03, D81 Version: April 2013 Keywords: prospect theory; loss aversion; utility for gains and losses; probability distortion; decision analysis; risk aversion * Corresponding author: Erasmus School of Economics, PO Box 1738, 3000 DR Rotterdam, the Netherlands. T: +31 10 408 1295, F: +31 10 408 9141. We gratefully acknowledge helpful comments from Peter P. Wakker and Horst Zank and financial support from the Erasmus Research Institute of Management, the Netherlands Organisation for Scientific Research (NWO), and the Tinbergen Institute. 2 1. Introduction An extensive amount of empirical work, originating from Ellsberg's (1961) famous thought experiment, shows that people are not neutral towards ambiguity, as assumed by subjective expected utility, but usually dislike ambiguity. New models have been proposed to explain this ambiguity aversion. Broadly speaking, these ambiguity models can be subdivided into two classes. The first class models ambiguity aversion through a difference in utility between risk and uncertainty. The best‐known of these models is the smooth ambiguity model of Klibanoff et al. (2005), which is increasingly popular in economic applications (e.g. Treich 2010, Gollier 2011). Other models that belong to this class were proposed by Nau (2006), Chew et al. (2008), Seo (2009), and Neilson (2010). The second class assumes that utility does not depend on the source of uncertainty and is the same for risk and uncertainty. Instead, ambiguity aversion is modeled through a difference in event weighting. This class includes the multiple priors models (Gilboa and Schmeidler 1989, Jaffray 1989, Ghirardato et al. 2004) and modifications thereof (Gajdos et al. 2008, Maccheroni et al. 2006), vector expected utility (Siniscalchi 2009), Choquet expected utility (Gilboa 1987, Schmeidler 1989), and prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992). This paper tests whether utility is the same for risk and uncertainty. Empirical evidence indicates that ambiguity attitudes differ between gains and losses (e.g. Cohen et al. 1987, Hogarth and Kunreuther 1989, Abdellaoui et al. 2005, Du and Budescu 2005) and that loss aversion is crucial in explaining attitudes towards both risk (Rabin 2000) and ambiguity (Roca et al. 2006). We, therefore, measured utility both for gains and for losses and we also measured loss aversion. We assume a general utility model, previously suggested by Miyamoto (1988), Luce (1991), and Ghirardato and Marinacci (2001) that includes all the models of the second class as special cases, and generalize it to include sign‐dependence. An empirical test supported the central condition underlying this model. Measuring loss aversion is complex, in particular if event weighting is allowed to differ for gains and losses, as we do. Previous measurements of loss aversion sidestepped this problem by introducing simplifying assumptions. We introduce a new method to measure loss aversion that imposes no simplifying assumptions. It can easily be applied, which may encourage the use of ambiguity models in practical decision problems where simple measurement methods are required. Our method extends the trade‐off method of Wakker and Deneffe (1996) by allowing standard sequences (sequences of outcomes for which the utility difference between successive elements is constant) to pass through the reference point. This makes it possible to measure utility on its entire domain and to quantify loss aversion. It provides a useful tool for practical applications of ambiguity models. It also simplifies the axiomatization of ambiguity models as there is a close connection between 3 measurements of utility using the trade‐off method and preference conditions (Köbberling and Wakker 2003). Our method provides a straightforward way to measure loss aversion without the need to fully measure utility. In particular, we provide an efficient way to operationalize Köbberling and Wakker’s (2005) measure of loss aversion. Our data support the assumption that utility and loss aversion are the same for risk and uncertainty. This suggests that utility primarily reflects attitudes towards outcomes. We also found evidence of ambiguity aversion for mixed acts. Taken together, these findings support models that explain ambiguity aversion through a difference in event weighting. This is the first message of our paper. The second message is that descriptive ambiguity models should allow for reference‐dependence. We obtained clear evidence that utility differed for gains and losses and there was sizeable loss aversion. Most ambiguity models do not allow for reference‐dependence and assume that ambiguity attitudes are the same for gains and losses. This assumption may be adequate for normative purposes, but not for descriptive purposes. 2. Background 2.1. Theory Consider a decision maker who has to make a choice in the face of uncertainty. Uncertainty is modeled through a state space . Exactly one of the states will obtain, but the decision maker does not know which one. Subsets of are called events and denotes the complement of . Acts map states to outcomes. Outcomes are money amounts and more money is preferred to less. In our measurements we will only need two‐outcome acts , signifying that the decision maker obtains € if event occurs and € otherwise. We use conventional notation to express the preference of the decision maker, letting ≻, ≽, and ∽ represent strict preference, weak preference, and indifference. Preferences are defined relative to a reference point . Gains are outcomes strictly preferred to and losses are outcomes strictly less preferred than . An act is mixed if it involves both a gain and a loss. For mixed acts the notation signifies that is a gain and is a loss. A gain act involves no losses (i.e. both and are nonnegative) and a loss act involves no gains. For gain and loss acts the notation signifies that the absolute value of exceeds the absolute value of , i.e. if and are gains then and if and are losses then . If probabilities are known, we will denote by the act that pays € with probability and € with probability 1. We will refer to as 4 an uncertain act (meaning that probabilities are unknown) and to as a risky act (meaning that probabilities are known). Under binary rank‐dependent utility the decision maker’s preferences over mixed acts are evaluated by: , (1a) and preferences over gain or loss acts by: 1 , (1b) where for gains and for losses. is a strictly increasing, real‐valued utility function that satisfies 0. is an overall utility function that includes loss aversion. In empirical application is often decomposed in a basic utility function, capturing the decision maker’s attitudes towards final outcomes, and a loss aversion coefficient , capturing attitudes towards gains and losses (Sugden 2003, Köbberling and Wakker 2005, Köszegi and Rabin 2006). Our method does not require this decomposition. The utility function is a ratio scale and we can choose the utility of one outcome other than the reference point. The event weighting functions , ,, assign a number to each event such that (i) ∅ 0 (ii) 1 (iii) is monotonic: ⊃ implies . The event weighting functions depend on the sign of the outcomes and may be different for gains and losses. They need not be additive. For gains, binary RDU contains most transitive ambiguity models as special cases, as was pointed out by Miyamoto (1988), Luce (1991), and Ghirardato and Marinacci (2001). The ambiguity models only differ when the number of outcomes is at least three. Equations (1a) and (1b) represent the extension of these models to include sign‐ dependence. Binary RDU evaluates mixed risky acts as 1 (2a) and gain and loss risky acts as 5 1 . (2b) is a strictly increasing probability weighting function that satisfies 0 0 and 1 1 and again may differ between gains and losses. Hence, in the evaluation of risky acts the event weighting functions are replaced by probability weighting functions . Binary RDU assumes that utility is the same for risk and uncertainty. Ambiguity aversion is modeled through a difference between and . 2.2. Previous evidence Tversky and Kahneman (1992) assumed that utility differs between gains and losses and is S‐shaped, concave for gains and convex for losses, and steeper for losses than for gains, reflecting loss aversion. Nearly all the empirical evidence on utility comes from decision under risk.
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